Generic extensions and multiplicative bases of quantum groups at 𝐪=0

Reineke, Markus

doi:10.1090/S1088-4165-01-00111-X

Generic extensions and multiplicative bases of quantum groups at ${{\mathbf q=0}}$
HTML articles powered by AMS MathViewer

by Markus Reineke

Represent. Theory 5 (2001), 147-163

DOI: https://doi.org/10.1090/S1088-4165-01-00111-X

Published electronically: June 12, 2001

PDF | Request permission

Abstract:

We show that the operation of taking generic extensions provides the set of isomorphism classes of representations of a quiver of Dynkin type with a monoid structure. Its monoid ring is isomorphic to the specialization at $q=0$ of Ringel’s Hall algebra. This provides the latter algebra with a multiplicatively closed basis. Using a crystal-type basis for a two-parameter quantum group, this multiplicative basis is related to Lusztig’s canonical basis.

References

Klaus Bongartz, On degenerations and extensions of finite-dimensional modules, Adv. Math. 121 (1996), no. 2, 245–287. MR 1402728, DOI 10.1006/aima.1996.0053
Klaus Bongartz, Some geometric aspects of representation theory, Algebras and modules, I (Trondheim, 1996) CMS Conf. Proc., vol. 23, Amer. Math. Soc., Providence, RI, 1998, pp. 1–27. MR 1648601
Jens Carsten Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1359532, DOI 10.1090/gsm/006
Daniel Krob and Jean-Yves Thibon, Noncommutative symmetric functions. V. A degenerate version of $U_q(\textrm {gl}_N)$, Internat. J. Algebra Comput. 9 (1999), no. 3-4, 405–430. Dedicated to the memory of Marcel-Paul Schützenberger. MR 1723475, DOI 10.1142/S0218196799000254
G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415, DOI 10.1090/S0894-0347-1990-1035415-6
George Lusztig, Cells in affine Weyl groups. II, J. Algebra 109 (1987), no. 2, 536–548. MR 902967, DOI 10.1016/0021-8693(87)90154-2
Markus Reineke, On the coloured graph structure of Lusztig’s canonical basis, Math. Ann. 307 (1997), no. 4, 705–723. MR 1464139, DOI 10.1007/s002080050058
Christine Riedtmann, Lie algebras generated by indecomposables, J. Algebra 170 (1994), no. 2, 526–546. MR 1302854, DOI 10.1006/jabr.1994.1351
Christine Riedtmann, Lie algebras generated by indecomposables, J. Algebra 170 (1994), no. 2, 526–546. MR 1302854, DOI 10.1006/jabr.1994.1351
Claus Michael Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583–591. MR 1062796, DOI 10.1007/BF01231516
Mitsuhiro Takeuchi, A two-parameter quantization of $\textrm {GL}(n)$ (summary), Proc. Japan Acad. Ser. A Math. Sci. 66 (1990), no. 5, 112–114. MR 1065785, DOI 10.3792/pjaa.66.112